Illinois Journal of Mathematics Volume 51, Number 4, Winter 2007, Pages 1341–1347 S 0019-2082
EXTREMAL CASES OF EXACTNESS CONSTANTS AND COMPLETELY BOUNDED PROJECTION CONSTANTS HUN HEE LEE
Abstract. We investigate some extremal cases of exactness constants and completely bounded projection constants. More precisely, for an √ n-dimensional operator space E we prove that λcb (E) = n if and only √ if ex(E) = n.
1. Introduction Exactness constants and completely bounded (c.b.) projection constants are fundamental quantities in operator space theory. For an operator space E ⊆ B(H), the c.b. projection constant of E, λcb (E), is defined by λcb (E) = inf{kP kcb | P : B(H) → E, projection onto E}. Let B = B(`2 ) and K be the ideal of all compact operators on `2 , and let TE : (B ⊗min E)/(K ⊗min E) → (B/K) ⊗min E be the map obtained from q ⊗ IE : B ⊗min E → (B/K) ⊗min E by the taking quotient with respect to K ⊗min E, where q : B → K is the canonical quotient map. Then the exactness constant of E, ex(E) is defined by
ex(E) = TE−1 . It is well known that the exactness constant is the same as dSK (E), where dSK (E) = inf {dcb (E, F ) : F ⊆ K} , when E is finite dimensional ([9]). The followings are well known facts about these quantities (Chapter 7 and 17 of [12] and Section 9 of [10]): Received May 4, 2006; received in final form June 19, 2006. 2000 Mathematics Subject Classification. Primary 47L25. Key words and phrases. Exactness constants, projection constants. Research supported by the BK21 program. c
2007 University of Illinois
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Fact 1.
HUN HEE LEE
For a finite dimensional operator space E we have ex(E) = dSK (E) ≤ λcb (E).
Fact 2.
When dim(E) = n ∈ N, we have √ λcb (E) ≤ n.
Thus, for √ an n-dimensional operator space E, λcb (E) and ex(E) are both bounded by n, and this upper bound is known to be asymptotically sharp. Indeed, we have ex(max `n1 ) ≥ 2√nn−1 for n ≥ 2 ([9]). However, it is not yet √ known whether √ there is an n-dimensional operator space E with λcb (E) = n or ex(E) = n. √ √ In this paper we investigate the extremal cases λcb (E) = n and ex(E) = n and prove the following theorem. Theorem 1. Let n ≥ 2 and operator √ E ⊆ B(H) be an n-dimensional √ space. Then we have λ (E) = n if and only if ex(E) = n. Equivalently, cb √ √ λcb (E) < n if and only if ex(E) < n. λcb (E) is the operator space analogue of the projection constant λ(X) of a Banach space X given by λ(X) = sup{λ(X, Y ) | X ⊆ Y }, where λ(X, Y ) = inf {kP k | P : Y → Y projection onto X} . See [4], [5], and [6] for more information on the Banach space case and [3] and [13] for the operator space case. Throughout this paper, we assume that the reader is familiar with the standard results about operator spaces ([2], [10]), completely nuclear maps ([2]), and completely p-summing maps ([11]). For a linear map T : E → F between operator spaces and 1 ≤ p < ∞ we denote the completely nuclear norm and the completely p-summing norm of T by ν o (T ) and πpo (T ), respectively. For an index set I, OH(I) denotes the operator Hilbert space on `2 (I), which was introduced in [10]. When I = {1, . . . , n} for n ∈ N, we simply write OHn . For a family of operator spaces (Ei )i∈I and an ultrafilter U on I Q we denote the ultraproduct of (Ei )i∈I with respect to U by U Ei . 2. Proof of the main result For the proof we need several lemmas. The first lemma is about the relationship between completely 1-summing maps and completely 2-summing maps. Lemma 2. Let v : E → F be a completely 1-summing map. Then v is completely 2-summing with π2o (v) ≤ π1o (v).
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Proof. Let E ⊆ B(H) for some Hilbert space H. Then by Remark 5.7 of [11] we have an ultrafilter U over an index set I and families of positive operators (aα )α∈I , (bα )α∈I , in the unit ball of S2 (H) such that the following diagram commutes for some u with kukcb ≤ π1o (v): v
E −−−−→ iy
(2.1)
F x u
E∞ −−−−→ E1 M
where E∞ = i(E) for the complete isometry Y i : B(H) ,→ B(H), x 7→ (x)α∈I , U
Q
E1 = M i(E) (the closure in U S1 (H)) for Y Y M: B(H) → S1 (H), (xα ) 7→ (aα xα bα ), U
U
and M = M |E∞ . Next, we split M into M = T2 T1 , where Y Y 1/2 T1 : B(H) → S2 (H), (xα ) 7→ (a1/2 α xα bα ) U
U
and T2 :
Y
S2 (H) →
U
Y
1/2 S1 (H), (xα ) 7→ (a1/2 α xα bα ).
U
Note that (2.2)
1/2 kT2 kcb ≤ lim Mα : S2 (H) → S1 (H) , x 7→ a1/2 α xbα U
≤ 1,
cb
since Mα∗ = Nα for 1/2 Nα : B(H) → S2 (H) , x 7→ a1/2 α xbα
and kNα kcb ≤ 1. Thus we have by Theorem 5.1 of [11] that k(vxij )kMn (F ) = k(uT2 T1 ixij )kMn (F ) ≤ π1o (v) k(T2 T1 ixij )kMn (S1 (H))
1/2 ≤ π1o (v) k(T1 ixij )kMn (S2 (H)) = π1o (v) (a1/2 x b )
ij α α
Mn (S2 (H))
for any n ∈ N and (xij ) ∈ Mn (F ), which implies π2o (v) ≤ π1o (v).
The second lemma is about the trace duality of completely 2-summing norms. Lemma 3. Let E and F be operator spaces and E be finite dimensional. Then for v : F → E we have (π2o )∗ (v) := sup {|tr(vu)| | π2o (u : E → F ) ≤ 1} = π2o (v).
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HUN HEE LEE
Proof. See Lemma 4.7 of [7].
The final lemma is about the relationship between the trace norm and the completely nuclear norm of a linear map on an operator space and the operator space approximation property. Lemma 4. Let E be an operator space with the operator space approximation property. Then for any completely nuclear map u : E → E we can define tr(u), the trace of u, and we have |tr(u)| ≤ ν o (u). Proof. Since E has the operator space approximation property, the canonical mapping b ∗ → E ⊗min E ∗ Φ : E ⊗E b (resp. ⊗min ) is the projective is one-to-one by Theorem 11.2.5 of [2], where ⊗ (resp. injective) tensor product in the category of operator spaces. Thus, N o (E), the set of all completely nuclear maps on E, can be identified with b ∗ with the same norm. Since we have the trace functional defined on E ⊗E b ∗ (7.1.12 of [2]), we can translate it to N o (E), so that we have E ⊗E o |tr(u)| ≤ kU kE ⊗E b ∗ = ν (u),
b ∗ is the element associated to u ∈ N o (E). where U ∈ E ⊗E
Let E and F be operator spaces. Then the Γ∞ -norm and the γ∞ -norm of a linear map v : E → F are defined by Γ∞ (v) = inf kαkcb kβkcb , where the infimum is taken over all Hilbert spaces H and all factorizations β
α
iF v : E → B(H) → F, where iF : F ,→ F ∗∗ is the canonical embedding, and γ∞ (v) = inf kαkcb kβkcb , where the infimum is taken over all m ∈ N and all factorizations α
β
v : E → Mm → F. See Section 4 of [3] or [1] for the details. Now we are ready to prove our main result. The proof follows the classical idea of [4]. Proof of Theorem√1. By Fact 1 and Fact 2 it is enough to show that the condition λcb (E) = n is inconsistent with the condition ex(E) = dSK (E) < √ n.
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√ Step 1. π1o (IE ) = n. By trace duality and Lemma 4.1 and 4.2 of [3] (or see Theorem 7.6 of [1]) we have |tr(u)| λcb (E) = Γ∞ (IE ) = γ∞ (IE ) = sup . o o u∈π1 (E) π1 (u) Since E is finite dimensional, we can find u ∈ CB(E) such that |tr(u)| √ = n, π1o (u) o and √ by multiplying by a suitable constant we can also assume that π2 (u) = n. Then, by Lemma 2, Lemma 3, and Theorem 6.13 of [11], we obtain √ √ n = nπ2o (u) ≤ nπ1o (u) = |tr(u)| ≤ π2o (u)π2o (IE ) = n.
Thus, we get
√ π1o (u) = n and |tr(u)| = n. Next, we show that u is actually IE . By Proposition 6.1 of [11] we have the factorization √ A B u : E → OHn → E with π2o (A) kBkcb ≤ n. If we let v : OHn → OHn be defined by v = AB, we have tr(v) = tr(v ∗ ) = tr(u) and 2 kIOHn − vkHS = tr (IOHn − v)(IOHn − v)∗ = tr(IOHn ) − 2 tr(u) + tr(vv ∗ ) 2
= n − 2n + kvkHS = (π2o (v))2 − n ≤ (π2o (A) kBkcb )2 − n ≤ 0, which leads to the desired conclusion. Step 2. Now we factorize IE as in the proof of Lemma 2. Then we have an ultrafilter U, families of positive operators (aα )α∈I , (bα )α∈I , in the unit ball of S2 (H), such that the diagram (2.1) commutes for some u with √ kukcb ≤ π1o (IE ) = n. Then we can find a rank n projection w1 : i(B(H)) → i(B(H)) onto E∞ with π1o (w1 ) ≤
√
n.
Consider iu : E1 → i(B(H)). Since i is a complete isometry, i(B(H)) is injective in the operator space sense, so that we can extend iu to Y u ˜: S1 (H) → i(B(H)) with k˜ ukcb = kiukcb . U
Now consider the same factorization M = T2 T1 as before. Note that π2o (T1 ) ≤ 1 and kT2 kcb ≤ 1
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HUN HEE LEE
by the same calculation as the proof for (5.8) of [11] and (2.2), respectively. Then for Y Y w := T1 u ˜ T2 : S2 (H) → S2 (H) U
U
we have (2.3)
kwkHS = π2o (w) ≤ π2o (T1 ) k˜ ukcb kT2 kcb ≤ π2o (T1 ) kukcb ≤
√
n.
Since T1 i is 1-1, F := T1 i(E) is n-dimensional. Furthermore, since wT1 ix = T1 u ˜T2 T1 ix = T1 iuMix = T1 ix for all x ∈ E, we have w|F = IF , which means that |λk (w)| ≥ 1 for 1 ≤ k ≤ n, where (λk (w))k≥1 is the sequence of eigenvalues of w, in non-increasing order and counted according to multiplicity. By applying Weyl’s inequality (Lemma 3.5.4 of [8]) and (2.3), we get n≤
n X k=1
2
|λk (w)| ≤
∞ X
2
sk (w)2 = kwkHS ≤ n,
k=1
where (sk (w))k≥1 is the sequence of singular values of w. Then we have ( 1 if 1 ≤ k ≤ n, |λk (w)| = 0 if k > n, which implies that w has rank at most n, as does w1 := u ˜M = u ˜T2 T1 |i(B(H)) : i(B(H)) → i(B(H)). Actually, w1 is our desired rank n projection. Indeed, we have w1 ix = u ˜Mix = iuMix = ix for all x ∈ E, and since E∞ is n-dimensional, w1 maps onto E∞ . Moreover, we have √ π o (w1 ) ≤ k˜ ukcb π1o (M) ≤ n since π1o (M) ≤ 1 ((5.7) of [11]). √ Step 3. Since dSK (E∞ ) = dSK (E) < n, we have F ∈ K and an isomorphism
√ T : E∞ → F with kT kcb T −1 cb < n. By the fundamental extension theorem (Theorem 1.6 of [12]) we have extensions −1 : B(` ) → i(B(H)) Te : i(B(H)) → B(`2 ) and Tg 2 −1 of T and T , respectively, with
−1
e
T = kT kcb and Tg
= T −1 cb . cb
cb
−1 : B(` ) → B(` ). Then clearly we have ran(w Let w˜1 = Tew1 Tg ˜1 ) ⊆ F and 2 2 w˜1 |F = IF , which means that w˜1 is also a rank n projection from B(`2 ) onto
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F . Since F ⊆ K and K satisfies the operator space approximation property, we have by Lemma 4 and Corollary 15.5.4 of [2] that n = |tr(w˜1 |K : K → K)| ≤ ν o (w˜1 |K : K → K) = π1o (w˜1 |K : K → K)
−1 = π1o (w˜1 |K : K → B(`2 )) ≤ Te Tg
π1o (w1 ) cb cb
−1 √
≤ kT kcb T n < n, cb This is a contradiction.
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[email protected];
[email protected]
